Lagrange's father was an officer in the French army cavalry. Later, his family was in decline due to business bankruptcy.According to the memory of Japanese Lagrangians, if he had a rich family when he was young, he would not do mathematical research because his father wanted to train him to be a manlawyer。Lagrange himself had no interest in law.
Lagrange's scientific research covers a wide range of fields.His most outstanding contribution in mathematics is to separate mathematical analysis from geometry and mechanics, making the independence of mathematics more clear. Since then, mathematics is no longer just a tool for other disciplines.
Joseph Lagrange
Lagrange summarized the mathematical achievements of the 18th centurynineteenth centuryMathematic research has opened the way, which can be called the most outstandingMathematician。At the same time, he talked about the movement of the moon（Three body problem）, planetary motionTrack calculation, two fixed centersfluid mechanicsSuch achievements have also played a historic role in making astronomy more scientific and mechanical analysis, promoted the further development of mechanics and celestial mechanics, and become pioneering or groundbreaking research in these fields.
stayBerlinIn the first ten years of his work, Lagrange spent a lot of timealgebraic equationandTranscendental equationHe has made valuable contributions to the solution of algebra and promoted the development of algebra.He submitted toBerlin Academy Two famous papers: "On solving numerical equations" and "Research on algebraic solution of equations".The various solutions to the third and fourth order algebraic equations of predecessors are summarized as a set of standard methods, that is, the equations are reduced to the lower order equations (called auxiliary equations orResolvent）To solve.
So does LagrangeAnalytical mechanicsFounder of.In his famous work Analytical Mechanics, Lagrange summarized the basic principles of various mechanics in history and developedDarumbel、EulerAnd others, introducing potential sumEquipotential surfaceAnd further apply mathematical analysis toparticleandRigid body mechanics, proposed the application ofstaticsanddynamicsThe general equation ofGeneralized coordinateThe concept ofLagrange equation, the mechanical systemEquation of motionThe change from the Newtonian form with force as the basic concept to the analytical mechanics form with energy as the basic concept has laid the foundation of analytical mechanics and opened the way for the application of mechanical theory to other fields of physics.
He also showed that the rigid body, under the action of gravityFixed-point rotationLagrangianEuler dynamic equationThe solution of the three body problem has an important contribution to the solution method of the three body problem, and has solved the stereotype problem of the restricted three body movement.Lagrange pairFluid motionThe theory ofLagrange method。
In Lagrange's research work, about halfCelestial mechanicsofHe used his principles and formulas in analytical mechanics to establish the equations of motion of various celestial bodies.In the solution of the equation of celestial motion, Lagrange found five special solutions of the equation of motion of the three body problem, namely the Lagrange translation solution.In addition, he also studiedcometandasteroidOfPerturbationThe comet origin hypothesis was proposed.
In the past hundred years, many new achievements in the field of mathematics can be traced directly or indirectly to the work of Lagrange.Therefore, in the history of mathematics, he is considered to be one of the mathematicians who have a comprehensive impact on the development of analytical mathematics.[1]
birth
Lagrange's father's surname is Lagrangia.The official name of Lagrange on the birth and baptism records in Turin is Giuseppe Lodovico, Lagrangian. His father's name is Francesco Lodovico, Lagrangian; his mother's name is Teresa Grosso.His surname used to be De la Grange, La Grange（La Grange）Etc.After his death, Joseph Louis Lagrange was officially used in the eulogy written to him by the French Academy.The father is of French descent.The great grandfather was a French cavalry colonelItalyLater, he married and settled down with the Roman family;My grandfather was the accountant of Turin's Public Affairs and Defense Bureau, and married the local people.His father also worked in the same unit in Turin and had 11 children, but most of them died prematurely, and Lagrange was the largest.
Youth
In his youth, under the guidance of the mathematician Revery, Lagrange lovedgeometry。When he was 17, he readbritainastronomerHarleyIntroduction to NewtonCalculusAfter his successful essay "On the Advantages of Analytical Methods", he felt that "analysis is his favorite subject". From then on, he became fascinated with mathematical analysis and began to specialize in the rapidly developing mathematical analysis at that time.
At the age of 18, Lagrangian daily useItalianI wrote my first paper with Newtonbinomial theoremWhen dealing with the higher derivative of the product of two functions, he usedLatinIt was written and sent toBerlin Academy MathematicianEuler。Soon after, he learned that this achievement had beenLeibnizYes.This unlucky beginning did not discourage Lagrange, on the contrary, it strengthened his confidence in the field of mathematical analysis.
Travel through the times
In 1755, when Lagrange was 19 years old, he was discussing mathematical problems“Isoperimetric problem”Based on Euler's ideas and results, he used pure analytical methods to find the extreme value of variation.The first paper, "Research on maximum and minimum methods", developed theVariational methodAnd laid a theoretical foundation for the variational method.The creation of the variational method made Lagrange famous in Turin, and at the age of 19, he became a professor of the Royal Artillery School of Turin, and was recognized as a first-class mathematician in Europe at that time.In 1756, on Euler's recommendation, Lagrange was appointedPrussiaCommunication academician of the Academy of Sciences.
German in 1766Frederick the Great When he sent an invitation to Lagrange, he said that there should be "the largest mathematician in Europe" in the court of "the largest king in Europe".So he was invited to Berlin and served as the director of the Mathematics Department of the Prussian Academy of Sciences. He lived for 20 years and began his heyday of scientific research in his life.During this period, he completed the book Analytical Mechanics, which isNewtonThe next importantclassical mechanicswork.The book uses variational principles and analytical methods to establish a complete and harmoniousMechanical system, which makes the mechanics analysis.In his preface, he declared that mechanics has become a branch of analysis.
In 1783, Lagrange's hometown established the "Turin Academy of Sciences", and he was appointed honorary president.After Frederick the Great died in 1786, he accepted the King of FranceLouis XVI's invitation to leave Berlin and settle downParis, until death.
During this period, he participated in the committee established by the Paris Academy of Sciences to study the unification of weights and measures in France, and served as the director of the French Metric Committee.In 1799, France was unifiedWeights and measuresWe have developed the internationally recognizedlength、the measure of area、volume、qualityLagrange has made great efforts to this end.
In 1791, Lagrange was electedRoyal SocietyMembers, successively inEcole Normale Superieure andParis Polytechnic SchoolProfessor of Mathematics.In 1795, the highest academic institution in France was established——French Research InstituteLater, Lagrange was elected Chairman of the Mathematical and Physical Committee of the Academy of Sciences.Since then, he has carried out research work again and compiled a number of important works: On the Solution of Numerical Equations of Any Order, Analytic Function Theory and Function Calculation Handout, summarizing a series of research work in that period, especially his own.
depart from the world for ever
On April 3, 1813,NapoleonHe was awarded the Grand Cross of the Empire, but Lagrange was bedridden. On the morning of April 11, Lagrange died.
Main contributions
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Lagrange has made great historical contributions to mathematics, mechanics and astronomy, but he is mainly a mathematician. The purpose of studying mechanics and astronomy is to show the power of mathematical analysis.More than 500 books, papers, academic report records and academic communications were published.
Lagrange's academic career was mainly in the second half of the 18th century.At that time, mathematics, physics and astronomy were the main subjects of natural science.The mainstream of mathematics is mathematical analysis developed from calculus, with the European continent as the center;The mainstream of physics is mechanics;The mainstream of astronomy is celestial mechanics.The development of mathematical analysis has deepened mechanics and celestial mechanics, and the topics of mechanics and celestial mechanics have become the driving force for the development of mathematical analysis.The representatives of natural science at that time made historic contributions to the three disciplines.The following are comments on Lagrange's main contributions.
mathematics
Pioneer of mathematical analysis
European mathematics after Newton and Leibniz split into two schools.Britain still adheres to Newton's geometric method in Mathematical Principles in Natural Philosophy, and the progress is slow;Continental EuropeLeibnizAnalytical methods created (includingalgebraic method）, making rapid progress, calledAnalytics（analysis）。Lagrange is the largest pioneer next to Euler, and has made pioneering contributions to the main branches founded in the 18th century.
Variational method
This is the field that Lagrange first studied, based on Euler's ideas and results, but starting from pure analytical methods, more perfect results can be obtained.His first paper, Recherches sur la m é thode minimais et minimies, is the prelude to his research on variational methods;Essai d'unenouvelle m é thode pour d é terminer les maxima et les minima defomules integrals ind é finds, published in 1760, is a representative work of using analytical methods to establish variational methods.When I wrote to Euler before publishing, I called the method in this paper "the method of variation".Euler affirmed and formally named this method "the calculus of variation" in his own paper.The branch of variational method is really established.
The Lagrange method is rightintegralCarry out extremalization,functionY=y (x) TBD.Instead of Euler and his predecessors changing the individual coordinates of the maximum or minimum curve, he introducedEndpointThe new curve y (x)+δ y (x) of (x1, y1), (x2, y2), δ y (x) is called theVariational。J correspondingincrementThe first and second order terms of △ J expanded by δ y, δ y 'are called primary variation δ J and secondary variation δ 2J.He used analytical methods to prove that the necessary condition for δ J to be zero isEuler equation。
He also continued to discuss the situation when the endpoint changes and twoindependent variableThe case of multiple integrals makes this branch continue to develop.After 1770, Lagrange also studiedIntegrand functionF containsHigher derivativeUnit weight sum ofMultiple integralIt has developed into the standard content of variational method.
differential equation
As early as the time of Turin, Lagrange had opposite coefficientordinary differential equation The research has made significant achievements.In the process of order reduction, he proposedAdjoint equationAnd prove that the adjoint equation of the nonhomogeneous linear variable coefficient equationAdjoint equationIs the original equationHomogeneous equation。He also extended Euler's results on homogeneous equations with constant coefficients to the case of variable coefficients, and proved that the general solution of homogeneous equations with variable coefficients can be obtained by multiplying some independent special solutions with arbitrary constants;And after knowing the m special solutions of the equation, the equation can be reduced in m valence.
During the Berlin period, his singular solutions to ordinary differential equations andSpecial solutionHe made a historic contribution, systematically studied the relationship between singular solutions and general solutions in the "Sur les int é graiesparticulieres des equations diff é rentielles" (1774), and clearly proposed the method of finding singular solutions from general solutions and their partial derivatives to integral constants to eliminate constants;It is also pointed out that the singular solution isEnvelope。Of course, his theory of singular solutions is not perfect, and the form of modern theory of singular solutions was completed by G. Darboux and others.
The study of ordinary differential equations was carried out in combination with the subject of celestial mechanics at that time.In Essai sur le probil é medes trois corps, completed by Lagrange in 1772, he found five special solutions of ordinary differential equations of three body motion: three are collinear cases of three bodies;Two are three bodies keeping equilateral triangles;It is called Lagrangian translation solution in celestial mechanics.He is withLaplaceA perfect arbitrary constant variation method plays an important role in the approximate solution of the equations of the multibody problem and promotes the establishment of the perturbation theory.
Lagrange isFirst order partial differential equationThe founder of the theory, Sur l'integration des é quationau differences partielles du premier order, completed in 1772, and General Integration Methods for First Order Linear Partial Differential Equations, completed in 1785(M é thode g é n è rale pourin t é grer les equations partielles du premier order loresque differences ne sont que lin è aires), systematically completed the theory and solution of first-order partial differential equations.
He first proposed that the solutions of first-order nonlinear partial differential equations can be classified into complete solutions, singular solutions, general integrals, etc., and gave the relationship between them.Later, it is further proved that the solution of the linear equation Pp+Qq=R (P, Q, R are functions of x, y, z) (5) is equivalent to the solution, and the solution (6) formula is equivalent to the solution of ordinary differential equations.(5) Equation is still called Lagrange equation.Interestingly, it can be seen from the above that first-order nonlinear partial differential equations can be transformed into ordinary differential equations.But Lagrange himself was not clear. When he solved a special first order partial differential equation in 1785, he said that he could not use this method. Maybe he forgot his results in 1772.In modern times, this method is sometimes called Lagrange method, also called Cauchy's characteristic method.Since Lagrange only discussed the case of two independent variables, he encountered difficulties in generalizing to n independent variables, which was later overcome by Cauchy in 1819.
Equation theory
Algebra in the 18th century belongs to analysis, and equation theory is an active field.In his first ten years in Berlin, Lagrange spent a lot of time on solving algebraic equations and transcendental equations.
He made historic contributions to the solution of algebraic equations.In the long paper R é flexions sur le resolution alg é brique equations (Complete Works III, pp 205-421), we summarized the various solutions of the predecessors to the third and fourth order algebraic equations as a set of standard methods, and also analyzed the reasons why the general third and fourth order equations can be solved by algebraic methods.The cubic equation has a quadratic auxiliary equation, whose solution is a function of the root of the cubic equation, and there are only two values under the replacement of the root;The solution of the auxiliary equation of the quartic equation has only three different values under the replacement of the root, so the auxiliary equation is a cubic equation.Lagrange called the solution of the auxiliary equation as the resolvent function (rational function) of the original equation root.He continued to search for the resolvent function of the equation of degree 5, hoping that this function was the solution of the equation of degree 5, but failed.Nevertheless, Lagrange's idea already contains the concept of permutation group, and permutations that keep the value of resolvent (rational) function unchanged form subgroups, whose order is a factor of the order of the original permutation group.So Lagrange is the pioneer of group theory.His thought was later NH. Abel and E. Galois adopted and developed, and finally solved the problem why general equations higher than the fourth degree cannot be solved by algebraic methods.
Lagrange also proposed a series solution of transcendental equation in 1770.Let p be an equation, which is the Lagrange series commonly used in celestial mechanics later.He did not discuss convergence himself, and Cauchy later worked out the convergence range of this series.
number theory
Lagrange began to study number theory at the beginning of his stay in Berlin. In his first paper, Sur la solution des probl é m è s in d é t è rmin é sdu seconde degr é s, and in Solution d'un probl è me d'arithmetique, which was sent to Turin's Analects, Euler discussed the Fermat equation x2-Ay2=1 (x,y. A is an integer), (9)
A more general Fermat equation is obtained in Nouvelle m é thode pour resoudveles probl è mes ind é temin é s en nombres entities
(B is also an integer) (10).The more extensive binary quadratic integral coefficient equation is also discussed.
, (11) and solved the integer solution problem.
Lagrange also proved another Fermat conjecture, "A positive integer can be expressed as the sum of up to four squares", which Euler has not solved for more than 40 years.In D é monostation d'un theorem nouveau concernant les nombres premiers published in 1773, the famous theorem was proved: the necessary and sufficient condition for n to be a prime number is (n-1)+1 is divisible by n.
Lagrange not only has a lot of achievements, but also has innovations in methods.For example, in the "Recherches d'arithm é tiques" (Collected Works III, pp. 695-795), the method and results used to study the solution of formula (11) are the basic literature of quadratic form theory.
Functions and infinite series
Like other mathematicians in the 18th century, Lagrange also believed that a function can be expanded into an infinite series, which is the generalization of polynomials.He also tried to establish the foundation of calculus with algebra.In his Analytic Function Theory...... (Anthology IX), the subtitle added to the title "contains the main theorems of differential calculus, and does not use the concepts of infinitesimal, or vanishing quantity, or limit and stream number, but boils down to the art of algebraic analysis", indicating his view.As the problem of limit and series convergence is avoided, it is impossible to establish real series theory and function theory, but some of their methods and results are still useful, and their views are also developing.
Lagrange got the mean value theorem of differential for the first time in Analytic Function Theory(Chapter VI of the book)F (b) - f (a)=f ′ (c) (b-a) (a ≤ c ≤ b.He also stressed that Taylor series can not be used without considering the remainder.Although he has not considered convergence, or even the existence of derivatives of all orders, he stressed that Rn should approach zero.It shows that he has noticed the problem of convergence.
His long-standing dispute with Euler and d'Alembert on whether an arbitrary function can be expressed as a trigonometric series has not been resolved, but has laid a foundation for the establishment of the theory of trigonometric series in the future.
Lagrange interpolation formula
Finally, I would like to mention that he proposed the famous Lagrange interpolation formula in the Basic Course of Mathematics in Normal Schools.
Until now, the computer is still in use when calculating a large number of midpoint interpolation.In addition, the Lagrangian arbitrary multiplier method in solving the relative minimax of multivariate functions and differential equations is also used today.
other
In addition to his pioneering contribution to the main branch of mathematical analysis established in the 18th century, he also began to pay attention to the problem of strictness.Although he evaded the concept of limit, he still admitted that calculus can be established on the basis of limit (Collected Works I, p.325). However, due to the lack of attention to strictness, the branch established is difficult to deepen at a certain stage.This may be the reason for his lack of research work in his later years.In his letter to Da Lambert on September 21, 1781, he said: "In my opinion, it seems that the (mathematical) mine has been dug very deep, and unless a new mine vein is found, it is bound to give it up..." (Collected Works X Ⅲ 368), which expressed his feelings and those of other colleagues.Facts show that mathematics developed more rapidly in the 19th century after the strict foundation of mathematical analysis was established.
Analytical mechanics
Founder of Analytical Mechanics
In his book Analytical Mechanics (1788), he absorbed and developed the research achievements of Euler and d'Alembert, and applied mathematical analysis to solve the mechanical problems of particles and particle systems (including rigid bodies and fluids).On the basis of summarizing various principles of statics, including the principle of virtual velocity established by him in 1764, he proposed the general principle of analysis statics, namely the principle of virtual work, andD'Alembert principleThe general equation of dynamics is obtained.For mechanical systems with constraints, he adopts appropriate transformations, introduces generalized coordinates, and obtains general equations of motion, that is, the first and second Lagrange equations.The whole book is written in the form of mathematical analysis, without a picture, so it is called Analytical Mechanics.The book also gives the basic theory of micro vibration near the equilibrium position of multi degree of freedom system, but it is not accurate to say that the vibration characteristic equation has multiple roots. This error was not corrected by K. Weierstrass (1858) and O.H. Sommerf (1859) until the middle of the 19th century.Lagrange studied the ideal fluid motion equation after Euler, and first proposed the concept of velocity potential and stream function, which became the basis of the theory of fluid irrotational motion.In Analytical Mechanics, he derived the fluid motion equation from the general equation of dynamics, focusing on fluid particles and describing the motion process of each fluid particle from beginning to end.This method is now called Lagrangian method to distinguish Euler method focusing on spatial points, but in fact this method has also been applied by Euler.Lagrange studied the rotation of an overweight rigid body at a fixed point and made a detailed analysis of the situation that the inertia ellipsoid of the rigid body is a rotating ellipsoid and the center of gravity is on the axis of symmetry.This case is called the Lagrangian case of a heavy rigid body.This research was not published before his death, and was collated by J. Binet and included in the appendix of the second edition of Fracture Mechanics (1818).Prior to this, Poisson obtained the same result independently in 1811.Lagrange also derived the equilibrium equation of elastic thin plate in 1811.[2]
Celestial mechanics
Founder of celestial mechanics
Celestial mechanics was born when Newton published the law of universal gravitation (1687), and soon became the mainstream of astronomy.Its subject content and basic theory were established in the late 18th century.The main founders are Euler, A.C. Clairaut, d'Alembert, Lagrange and Laplace.Finally, the classical celestial mechanics was formally established by Laplace.About half of Lagrange's research work in his life was related to celestial mechanics, but he was mainly a mathematician. He wanted to treat mechanics as a branch of mathematical analysis and celestial mechanics as a branch of mechanics.Nevertheless, he still made significant historical contributions to the founding of celestial mechanics.
First of all, in establishing the equations of motion of celestial bodies, Lagrange used his principles in analytical mechanics and equations (16) and (17) to establish the equations of motion of various celestial bodies.In particular, according to his method of arbitrary constant variation in the solution of differential equations, he established the equation of motion with the number of elliptical orbits of celestial bodies as the basic variable, which is still called the Lagrange equation of planetary motion, and is widely used. This equation has played an important role in the establishment and improvement of the perturbation theory,The equation was given in the paper Recherches sur lath é orie des perturbations queles com è tes peuvent é prouver par l'action des plan è tes, which won the prize of the Paris Academy of Sciences in 1780, and was highly praised by d'Alembert and Laplace.In addition, in an award-winning article on the three body problem, the equations of motion of the three body problem were reduced to seven orders for the first time.
In the solution of the equation of celestial motion, Lagrange's major historical contribution is to find five special solutions of the equation of motion of the three body problem, namely, Lagrange translational solutions.Two of the solutions are that the three bodies always keep equilateral triangles in the process of elliptical movement around the center of mass.His theory was confirmed more than 100 years later.On February 22, 1907, the Heidelberg Observatory in Germany discovered an asteroid, which was later namedMYTHOSInHerculesAchilles(Achilles), No. 588, whose position coincides with the formation of the sun and JupiterEquilateral triangle。By 1970, 15 such asteroids had been discovered, all named after the generals in the Trojan War in Greek mythology.There are nine near the Lagrangian solution 60 ° ahead of Jupiter's orbit, called the Greek group;There are six near the solution at 60 ° behind Jupiter's orbit, called Trojan Group.After 1970, more than 40 asteroids were found in these two groups, including ChinaPurple mountain observatoryFour were found, but not named.As for the reason why there are still asteroids near the special solution, it is because these two special solutions are stable.In 1961Lunar orbitThe discovery of meteor matter gathered at the equilateral triangle solution with the Earth and the Moon before and after is another proof of Lagrange's solution.Up to now, no objects that must be near the three Lagrangian collinear groups (three body collinear case) have been found, because these three special solutions are unstable.In addition, Lagrange also made important contributions to the first-order perturbation theory, proposed a method to calculate long-term perturbation (Collection V, pp. 125-414), and proposed the stability theorem of the solar system under the first-order perturbation together with Laplace(See "Laplace" in Biography of World Famous Scientists Astronomer I)。In addition, the Lagrange series (8) formula is widely used in perturbation theory.
Lagrange also made a lot of important contributions to the research on the movement of specific celestial bodies, most of which were the subjects of the prize collection of the Paris Academy of Sciences.His research papers on the theory of lunar motion have won many awards.Recherches sur laLibration de la lune, completed in 1763, won the 1764 annual award. This paper explains the angular velocity difference between the rotation and revolution of the moon well, but does not explain the rotation laws of the moon's equator and orbital plane well enough.Later, the paper completed in 1780 was solved better (see Anthology V, pp. 5-123). The winner of the 1772 annual award was the famous paper on the three body problem, which was also written for the study of the moon movement.The paper that won the 1774 annual prize was "Sur l'equation s é culaire de la lune", in which the shape of the earth and the perturbations of all major planets on the moon were discussed for the first time.The paper on the motion of planets and comets also won two prizes.In 1776, he won the prize in three papers completed in 1775, in which he discussed the impact of long-term changes in the intersection and inclination of planetary orbits on the motion of comets.The award-winning paper in 1780 was the one that proposed the famous Lagrange equation of planetary motion.The paper that won the 1766 annual prize was "Recherches sur les in é gualit é s des satellites de Jupiter...", in which the influence of solar gravity on the motion of Jupiter's four satellites was discussed for the first time, and the result was better than that of d'Alembert.
Lagrange is still engaged in many celestial mechanics topics. For example, in the first half of the Berlin period, he also studied the method of calculating the comet orbit with the observation data of three times (Anthology IV, pp. 439-532), and the results obtained become the basis of orbit calculation.In addition, he also obtained a mechanical model - the solution of the problem of two fixed centers, which was discussed by Euler, also known as the Euler problem.It is Lagrange's generalization to the case where centrifugal force exists, so it is later called Lagrange's problem (Collected Works II, pp. 67-121). These models are still in use.It is used as an approximate mechanical model for the motion of artificial satellites.In addition, the results of hydrostatics given by him in Analytical Mechanics later became the basis for discussing the theory of celestial body shape.
On the whole, Lagrange's historical contribution among the five founders of celestial mechanics is second only to that of Laplace.His "analytical mechanics" has a profound influence on the development of celestial mechanics.
Research experience
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Lagrange completed a large number of major research achievements during his stay in Berlin, which was the peak of his life's research. Most of his papers were published in the above two journals, and a small number were still sent back to Turin.Among them, the achievements in lunar motion (three body problem), planetary motion, orbit calculation, two fixed centers problem, fluid mechanics, number theory, equation theory, differential equation, function theory, etc. have become pioneering or groundbreaking research in these fields.In addition, he has also made important contributions to probability theory, cyclic series, and some mechanics and geometry topics.He also translated the works of Euler and A. De Moivre.In a letter to P. Laplace in 1782, he said: "I have almost finished writing Trait é de M é canique Analytice, but it cannot be published." Laplace arranged to publish the book in Paris. It was already 1788 when the book was published, and Lagrange had already arrived in Paris.This book became the foundation work of analytical mechanics.
In 1783, the "Turin Academy of Sciences" was established in his hometown, and Lagrange was appointed honorary president.The original publication was changed to M é langes des l'Acade mie des sciences des Turin.Lagrange often sends papers back for publication.In August 1786, Frederick II, the Prussian king who supported him, died and decided to leave Berlin.He left for France on May 18, 1787, at the invitation of the Paris Academy of Sciences.
Paris period (1787-1813).Lagrange officially worked at the Paris Academy of Sciences on July 29, 1787.Since he has been the associate academician of the Academy since 1772, his work was welcomed more warmly this time, but it was a pity that Dalanberg died in 1783.
In his first few years in Paris, he mainly studied more extensive knowledge, such as metaphysics, history, religion, medicine and botany.When the bourgeois revolution broke out in 1789, he just looked on with interest.On May 8, 1790, the Constitutional Assembly passed the decimal metric system law, and the Academy of Sciences established a corresponding "measurement committee", with Lagrange as one of its members.On August 8, the National Assembly decided to exercise dictatorship over the Academy of Sciences. Three months later, it decided to put A50. Lavoisier, Laplace, C. A. Coulomb and other famous academicians were eliminated from the Academy of Sciences.But Lagrange was retained and served as chairman of the Weights and Measures Committee.
In 1792, Lagrange, who had been widowed for nine years, married Ren é e-Francoise Adelaide, daughter of astronomer LeMonnier. Although he had no children, his family was happy.
In September 1793, the government decided to arrest all people born in the enemy countryLavoisierAfter trying to explain to the authorities, Lagrange was taken as an exception.
On May 7, 1794, the French Jacobins held a court session to try the figures of the tax organization of the Bourbon Dynasty, and sentenced all 28 members, including Lavoisier, to death. Lagrange and others tried their best to save them and asked for pardon, but they were rejected by J.B. Coffinhal, the Deputy Chief of the Revolutionary Court,He declared: "The Republic does not need scholars, but only just actions for the country!"
On the morning of May 8 the next day, Lagrange said sadly: "They can cut off Lavoisier's head in the blink of an eye, but his mind will never grow again in a hundred years."
In 1795, the National Longitude Bureau was established to uniformly manage the National Commission for Navigation, Astronomical Research and Weights and Measures. Lagrange was one of the members.Among the two French top universities, the Normal School and the Comprehensive Engineering School, established in the same year, Lagrange and others were among the first professors.After the abolition of the dictatorship over the Academy of Sciences, the French Academy, the highest academic institution in France, was established in 1795, and Lagrange was elected as the chairman of the mathematical and physical committee of the first branch (namely, the Academy of Sciences).Since then, he has only carried out research again, but mainly sorted out the past work and compiled a number of important works in combination with the teaching materials.
After the publication of "A Treatise on Analytical Mechanics" in 1788, Lagrange began to extend the principles and methods in the book to general situations.Some of his papers published before 1810,
For example, Memoirs surla th é orie g é n è rale de la variable des constants arbitrairesdans tons les pro bl è mes de la m é canique (read out in March 1809) published in Memoires de l 'Institute are all in preparation for the revision of the second edition.The second edition was renamed M é - canique analytic, which is divided into two volumes. The first volume was published in 1811, and the second volume was not printed until 1816. Lagrange has died for three years.
His textbook, Les le cons é l è mentales sur les Math é matique donn é s à l 'cole Normal, was published in 1796, and later included in the Oeuvres de Lagrange (hereinafter referred to as "Collected Works"). He enriched the contents of the seventh volume in 1812.
Train é de la r é solution des é qnations num é riques de tous les degr é s, published in 1798, summarized early achievements in equation theory, systematized them, and republished them in 1808 after enrichment.
He has published two historical works on function theory.The first is the Theory of Analytic Functions, which contains the main theorem of differential calculus. It does not use the concepts of infinitesimal, vanishing quantity, or limit and stream number, but sweeps into the art of algebraic analysis,De limits et de fluxions, et r é duits à l'analyse alg é brique de quantit é s finds), published in 1797 and reprinted in 1813;Another book, Lessons sur le calculus des functions, was published in 1801 and adapted from the handouts of teachers' schools.
1799“Mist Moon Coup”Later, Napoleon nominated Lagrange and other famous scientists asThe House of LordsMembers and members of the newly established Honorary Legion of the Order were appointed Earls;He was also awarded the Grand Cross of the Empire on April 3, 1813.By this time, Lagrange was seriously ill and finally died on the morning of April 11.At the funeral, President Laplace represented the House of Lords, and President Lac é p è de delivered a eulogy on behalf of the French Academy.All Italian universities held commemorative activities, but no activities were held in Berlin because Prussia joined the anti French alliance at that time.
Character works
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Lagrange's works are too many to be collected completely.After his death, the French Academy collected all the works he left in the Academy, edited and published fourteen volumes of Lagrange Anthology, which was edited by J.A. Serret. The first volume was published in 1867, and the fourteenth volume was printed in 1892.The first volume collects his work in Turin, published in the first to fourth volumes of the Analects;The second volume collects his papers published in the fourth and fifth volumes of the Analects and the first and second volumes of the Literature of the Turin Academy of Sciences;The third volume contains his papers published in the Literature of the Berlin Academy of Sciences (1768-1769, 1770-1773);The fourth volume has his papers published in New Literature of the Berlin Academy of Sciences (1774-1779, 1781, 1783);The fifth volume carries the papers published in the above publications (1780-1783, 1785-1786, 1792, 1793, 1803);The sixth volume contains his articles that have not been published in the publications of the Paris Academy of Sciences or the French Academy of Sciences;The seventh volume mainly publishes his report in the Normal School;The eighth volume is the book Trait é des é equations num é riquesde TOUS les degr é s, avec des notes sur plusieurs points de lath é orie des equations algo é briques, completed in 1808;The ninth volume is the book Analytic Function Theory, which was republished in 1813 and contains the main theorems of differential calculus. It does not use the concepts of infinitesimal, or vanishing quantity, or limit and stream number, but boils down to the art of algebraic analysis;The tenth volume is the book "A Course in Function Calculation" published in 1806;The eleventh volume is the first volume of Analytical Mechanics published in 1811 and annotated by J. Bertrand and G. Darboux;The twelfth volume is the second volume of Analytical Mechanics, which is still annotated by the above two people, and was later reprinted in Paris (1965);Volume 13 publishes his academic correspondence with D'Alembert;The fourteenth volume is his academic correspondence with Condorce, Laplace, Euler and others, and these two volumes are annotated by L. Lalanne.Volume 15, containing newsletters found after 1892, was also planned, but not published.[1]
Character evaluation
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Lagrange iseighteenth centuryHe has made historic contributions to mathematics, mechanics and astronomy.But his achievements are mainly in the field of mathematics,NapoleonHe once praised him as "a pyramid towering in the mathematical world", and his most outstanding contribution is tomathematical analysis The foundation ofgeometryAnd mechanics played a decisive role.Make the independence of mathematics more clear, not only as a tool for other disciplines.At the same time, it has also played a historic role in making astronomy more scientific and mechanical analysis, promoting mechanics and astronomy（Celestial mechanics）Further development.Due to historical limitations, his lack of rigor prevented him from achieving more results.
